text large_stringlengths 384 2.05k | rank_avg float64 1 4.19k ⌀ | rank_max float64 1 8.21k ⌀ | rank_min float64 1 5.03k ⌀ | rank_median float64 1 4.21k ⌀ | rank_by_avgsim float64 1 4.19k ⌀ | avgsim_to_github float32 0.77 0.85 ⌀ | dataset large_stringclasses 1
value |
|---|---|---|---|---|---|---|---|
ac{1}{k} \sum_{j=1}^k {\mathbb{E}}\left({\left\Vert X_j \right\Vert}^2
{\mathbbm{1}_{{\left\Vert X_j \right\Vert} > {\varepsilon}\sqrt{k}}} \right).$$ Then $$d_{BL} \left(\frac{1}{\sqrt{k}} \sum_{j=1}^k X_j, Z\right) \le
C_d \inf_{0 \le {\varepsilon}\le 1} ({\varepsilon}+ \theta({\varepsilon})),$$ where $Z$ is ... | 2,801 | 4,533 | 2,162 | 2,132 | null | null | github_plus_top10pct_by_avg |
$\Phi(\mathbf{x})$ comes from the background density $\rho_{\hbox{\scriptsize{asymp}}}$. Thus, a good fraction of the mass in the observable galaxy *does not contribute to the motion of test particles in the galaxy*. It is rather the near-core density $\rho_{II}^1(r)$ that contributes to $\mathfrak{V}(\mathbf{x})$. As ... | 2,802 | 4,441 | 2,524 | 2,726 | 2,533 | 0.779095 | github_plus_top10pct_by_avg |
turbation theory if the superdimension of the representation of the primary is non-zero (i.e. for short multiplets). For example for the short, discrete representation crucial to the calculation in [@Ashok:2009jw], there are no corrections.
Notice that the stress-energy tensor can also be written in terms of the right... | 2,803 | 1,451 | 1,035 | 2,599 | 3,911 | 0.769333 | github_plus_top10pct_by_avg |
uspin-DeltaACP.bib'
title: 'The Emergence of the $\Delta U=0$ Rule in Charm Physics'
---
Introduction \[sec:intro\]
==========================
In a recent spectacular result, LHCb discovered direct CP violation in charm decays at 5.3$\sigma$ [@Aaij:2019kcg]. The new world average of the difference of CP asymmetries [... | 2,804 | 984 | 2,632 | 2,785 | 1,055 | 0.795112 | github_plus_top10pct_by_avg |
cal{U}}(\chi )_{{\alpha }_i}$, and $F_i\in {\mathcal{U}}(\chi )_{-{\alpha }_i}$ for all $i\in I$. Let $${\mathbb{N}}_0^I=\Big\{\sum _{i\in I}a_i{\alpha }_i\,|\,a_i\in {\mathbb{N}}_0\Big\}\subset
{\mathbb{Z}}^I,$$ and for any subspace ${\mathcal{U}}'\subset {\mathcal{U}}(\chi )$ and any $\beta \in {\mathbb{Z}}^I$ let... | 2,805 | 1,752 | 2,782 | 2,560 | null | null | github_plus_top10pct_by_avg |
{\mbox{\boldmath $\alpha$}})
= \oplus_{P \in {\mathbb T}({\mbox{\boldmath $\alpha$}}) }K_0 v_P$ be a vector space over $K_0$ with the standard basis $\{v_P|P\in {\mathbb T}({\mbox{\boldmath $\alpha$}})\}$.
For generators $e_i$, $f_i$ and $s_i$ of ${A}_n$, we define linear maps $\rho_{{\mbox{\boldmath $\alpha$}}}(e_i)$... | 2,806 | 1,786 | 1,987 | 2,720 | 3,323 | 0.773165 | github_plus_top10pct_by_avg |
}A\quad & \textrm{Equation}\!:\textrm{ }{\textstyle {\mu\Phi_{x}+\Phi=(c/2)\int_{-1}^{1}\Phi(x,\mu^{\prime})d\mu^{\prime},\; x\geq0}}\\
& \textrm{Boundary condition}:\textrm{ }\Phi(0,\mu)=0,\;\mu\geq0\\
& \textrm{Asymptotic condition}:\textrm{ }\Phi\rightarrow e^{-x/\nu_{0}}\phi(\mu,\nu_{0}),\; x\rightarrow\infty.\\
... | 2,807 | 3,416 | 3,498 | 2,738 | 3,610 | 0.771238 | github_plus_top10pct_by_avg |
plest examples would be $$\begin{aligned}
{\label{eq:Juniform-def}}
J_{o,x}=\frac{{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{0<\|x\|_\infty\leq L\}$}}}}{\sum_{z\in{{\mathbb Z}^d}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{0<\|z
\|_\infty\leq L\}$}}}}=O(L^{-d})\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{0<\|L^{-... | 2,808 | 1,880 | 1,355 | 2,879 | null | null | github_plus_top10pct_by_avg |
the intrinsic fluctuations of the outflow rates present in the quasi-steady states.
![The net accretion rate $\dot{M}$ normalized by the Bondi rate $\dot{M}_{\mathrm{B}}$ as a function of the opening angle of the horizontal neutral layer at the Bondi radius $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$. The crosses show ... | 2,809 | 538 | 3,250 | 2,978 | 3,590 | 0.771336 | github_plus_top10pct_by_avg |
i \setminus \Phi][\Phi]$ is well-defined. By definition \[D:DYADIC\_ENSEMBLE\_PRODUCT\], $[\Psi \setminus \Phi][\Phi] = (\Psi \setminus \Phi) \cup \Phi = \Psi$.
The case $\Phi = \varnothing$ does not occur naturally in systems theory because no proper system is unresponsive to all possible stimuli. When $\Psi = \Phi$,... | 2,810 | 1,073 | 1,441 | 2,636 | 1,381 | 0.790286 | github_plus_top10pct_by_avg |
Therefore, using the fact that $(e_i - e_{\i})(e_i - e_{\i})^\top$ is positive semi-definite, and Equations , and we have $$\begin{aligned}
\label{eq:topl_expec}
\E[M] &\succeq& e^{-2b} \sum_{j = 1}^n \frac{\ell_j}{\kappa_j(\kappa_j-1)} \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i - e_{\i})^\top = e^{-2b} L,\end{aligned}... | 2,811 | 1,121 | 1,228 | 2,796 | null | null | github_plus_top10pct_by_avg |
roject2).
I am also taking AI this semester and was considering expanding upon something
like this for my masters' capstone project. The application of machine
learning to this type of situation is going to be very interesting to say the
least.
Just wanted to say good luck!
------
juskrey
Basically optimizations and... | 2,812 | 2,178 | 525 | 2,630 | 886 | 0.798318 | github_plus_top10pct_by_avg |
$i\in I$, $a\in A$.
3. ${\sigma }_i^a(R^a) = R^{{r}_i(a)}$ for all $i\in I$, $a\in A$.
4. If $i,j\in I$ and $a\in A$ such that $i\not=j$ and $m_{i,j}^a$ is finite, then $({r}_i{r}_j)^{m_{i,j}^a}(a)=a$.
If ${\mathcal{R}}$ is a root system of type ${\mathcal{C}}$, then ${\mathcal{W}}({\mathcal{R}})={\mathcal{W}}({\... | 2,813 | 1,138 | 2,122 | 2,687 | null | null | github_plus_top10pct_by_avg |
c_\gamma e^{-\gamma}$ with $\Gamma_G \subseteq \Lambda^*_G$ finite and all $c_\gamma \neq 0$. Then, $$m_{G,V}(\lambda) = \sum_{\gamma \in \Gamma_G} c_\gamma \, m_{T_G,V}(\lambda + \gamma).$$
In particular, it is evident from that, for any fixed group $G$, the multiplicity of an irreducible representation in some repre... | 2,814 | 3,490 | 2,507 | 2,512 | null | null | github_plus_top10pct_by_avg |
- Q^TG^{-1} Q + e^T e \ ,$$ where we introduce the frame fields $$G= \kappa^T \kappa \ , \quad e= \kappa \left( L + G^{-1} Q \right) .$$
We perform the dualisation by introducing a $\mathfrak{h}$-valued connection with components $A_{\pm} = A_{\pm}^{a}H_{a}$ and a $\mathfrak{h}^\star$-valued Lagrange multiplier $V= v_... | 2,815 | 1,891 | 2,557 | 2,572 | null | null | github_plus_top10pct_by_avg |
athbf{b}_j^T\mathbf{BX}^\dagger\bigg)^T$$ $$=\frac{1}{\lambda_i\lambda_j}\mathbf{b}_i^T(\mathbf{BX}^\dagger)(\mathbf{BX}^\dagger)^T\mathbf{b}_j=\frac{1}{\lambda_i\lambda_j}\mathbf{b}_i^T\mathbf{B}\mathbf{b}_j$$ $$=\frac{1}{\lambda_i}\mathbf{b}_i^T\mathbf{b}_j=0,$$ so $$\mathbf{c}_i^T\mathbf{c}_j=\frac{\mathbf{m}_i^T\ma... | 2,816 | 3,655 | 2,534 | 2,594 | null | null | github_plus_top10pct_by_avg |
owth condition]{}**]{}).
This is the basis of the following characterization theorem. For the proof we refer to [@PS91; @Kon80; @HKPS93; @KLPSW96].
\[charthm\] A mapping $F:S_{d}({\mathbb{R}}) \to {\mathbb{C}}$ is the $T$-transform of an element in $(S)'$ if and only if it is a U-functional.
Theorem \[charthm\] enab... | 2,817 | 1,739 | 2,102 | 2,531 | null | null | github_plus_top10pct_by_avg |
4(q)$, $|T_1|$ divides $q^6-1$), for all groups except ${}^2B_2(q), {}^2G_2(q), {}^2F_4(q) $. In the latter cases, denoted by $(\star)$, the information can be obtained from [@VV2 Lemma 2.8]
The previous lemmas provide the following result on the center of the prime graph of a simple group of Lie type, which can also ... | 2,818 | 1,634 | 1,528 | 2,610 | 3,632 | 0.771135 | github_plus_top10pct_by_avg |
You only need to annotate the type of an empty list. Lists with at least one
element don't require a type annotation because the type can be inferred from
the type of that element.
Dhall does not have buit-in support for homogeneous maps. Dhall does have
statically typed heterogeneous records (i.e. something like `{ ... | 2,819 | 1,006 | 1,676 | 2,061 | 1,095 | 0.794442 | github_plus_top10pct_by_avg |
eraction, i.e. they develop bulge rotation, bars and spiral arms in the first few hundred Myr of the simulation.
[^2]: We should note, however, that most $2\sigma$-galaxies do not exhibit a centrally peaked velocity dispersion, like the one presented here.
[^3]: The suggested mechanism could be responsible for the fo... | 2,820 | 394 | 2,863 | 2,650 | null | null | github_plus_top10pct_by_avg |
s used to estimate value of $\lambda$ by applying Hill’s method [@gopi.personal]. The choice $p=0.03$ provides results in line with Ref. [@gopi.volume], for $\Delta t = 15$ min time windows one finds $\lambda = 1.67 \pm 0.20$. There are several issues with this approach:
1. $p$ is a parameter that can be chosen arbit... | 2,821 | 2,494 | 4,173 | 2,752 | null | null | github_plus_top10pct_by_avg |
der. Pick $g_1,g_2,g_3\in G$ and let us check that $D(g_1H,g_3H)\leq D(g_1H,g_2H)+D(g_2H,g_3H)$. Choose an arbitrary $\varepsilon>0$ and and some $h_1,h_2, h'_2,h_3\in H$ such that $D(g_1H,g_2H)\geq d(g_1h_1,g_2h_2)-\varepsilon$ and $D(g_2H,g_3H)\geq d(g_2h'_2,g_3h_3)-\varepsilon$. Then $$D(g_1H,g_3H)\leq d(g_1h_1,g_3h... | 2,822 | 2,874 | 2,822 | 2,541 | null | null | github_plus_top10pct_by_avg |
s statement implies Theorem \[main\].
The first part of Theorem \[mainmain\] has been established above. In order to prove the second part, we will define a simple notion of ‘equivalence’ of germs (Definition \[equivgermsnew\]), such that, in particular, equivalent germs $\alpha(t)$ lead to the same component of the P... | 2,823 | 2,340 | 3,018 | 2,660 | 3,184 | 0.774207 | github_plus_top10pct_by_avg |
to their results.
This paper is organized as follows. In Section 1 we will prove Theorem \[main\] and derive Corollaries \[bgln\] and \[string\]. In Section 2 we describe the $K$-theoretic implications of Thoerem \[main\], and in particular we prove Theorem \[ktheory\].
Automorphisms of $R$-module bundles
===========... | 2,824 | 1,368 | 2,637 | 2,786 | 3,484 | 0.772024 | github_plus_top10pct_by_avg |
values. (**A**) Hazelnut tested in 1 µL of sample in 99 µL of running buffer (RB) (**B**) Peanut tested in 1 µL of sample in 99 µL of RB. (**C**) Hazelnut tested in 25 µL sample in 75 µL of RB. (**D**) Peanut tested in 25 µL sample in 75 µL of RB. (**E**) Hazelnut tested in 75 µL sample in 25 µL of RB. (**F**) Peanut t... | 2,825 | 147 | 3,058 | 3,199 | null | null | github_plus_top10pct_by_avg |
`VD+LDVYSDAY`
`VDLLDVYSDAY` ... | 2,826 | 6,953 | 552 | 715 | null | null | github_plus_top10pct_by_avg |
gle\left [\xi_{\rm eq}(t)
+g'(x)\xi_{\rm neq}(t)\right ]\left [\frac{\partial}{\partial v^{-\tau}}\{
\xi_{\rm eq}(t-\tau)+g'(x^{-\tau})\xi_{\rm neq}(t-\tau)\}\right]\rangle
p \hspace{0.2cm},\end{aligned}$$ where we have used the fact that the Jacobian obey the equation $^{12}$ $$\frac{d}{dt}\log\left|\frac{d(x^{t},v^{t... | 2,827 | 2,666 | 2,695 | 2,852 | null | null | github_plus_top10pct_by_avg |
} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K m}
\left\{ (UX)^{\dagger} A W \right\}_{m L}
\left\{ W ^{\dagger} A (UX) \right\}_{L l}
\biggr\}.
\label{P-beta-alpha-W4-H4-double}\end{aligned}$$
$$\begin{aligned}
&& P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{5th-1st}
\equiv
2 \mbox{Re} \left[ \left(... | 2,828 | 2,646 | 2,551 | 2,805 | null | null | github_plus_top10pct_by_avg |
}\cdot {\ensuremath{\operatorname{\mathbf{Pr}}\left[i\in D_t\right]}}.
\end{aligned}$$ In order to have $i\in D_t$, first an edge containing $i$ must be selected, and then the chosen $d$-element subset of that edge must contain $i$. By the $\beta$-balancedness property, $${\ensuremath{\operatorname{\mathbf{Pr}}\lef... | 2,829 | 1,363 | 2,330 | 2,600 | null | null | github_plus_top10pct_by_avg |
------------------------------------------------------------------------
Figure \[fig::confint\] shows typical confidence intervals for the projection parameter, $\beta_{{\widehat{S}}}$, and the LOCO parameter, $\gamma_{{\widehat{S}}}$, for one realization of each Setting. Notice that confidence intervals are only con... | 2,830 | 1,489 | 1,067 | 2,508 | 989 | 0.796219 | github_plus_top10pct_by_avg |
imply because the associated graded ring of $U_c$ is ${\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]^{{W}}$. The results we need follow easily from the corresponding results on $\operatorname{Hilb^n{\mathbb{C}}^2}$ and so we begin with the latter.
The (isospectral) Hilbert scheme {#isospecsec}
-------------------... | 2,831 | 2,058 | 1,550 | 2,698 | null | null | github_plus_top10pct_by_avg |
{h}}{ \,{}_{^{^{\bullet}}}}b_2)$ and $[{\mathbf{E}}, b_1b_2] = [{\mathbf{E}},b_1]b_2 + b_1[{\mathbf{E}},b_2]$. By induction, it therefore suffices to prove the result when $b=em\delta e\in B_{k,k-1}=
eH_{c+k} \delta e$, for some $k>0$. By we see that ${\mathbf{h}}{ \,{}_{^{^{\bullet}}}}b = e[{\mathbf{h}}_{c+k}, m]\del... | 2,832 | 1,853 | 1,500 | 2,854 | null | null | github_plus_top10pct_by_avg |
_k+e_l+\Gamma_i, & \mbox{if} & k\neq l, \end{array} \right.$$ Thus $$\Gamma(a_{F_i})\setminus\Gamma(a_{F_1},\ldots,a_{F_{i-1}})=\Union_{L\in\mathcal L}(e_L+\Gamma_i),$$ where $${\mathcal L}=\{\{k_1,\ldots, k_{i-1}\}\: k_j\in F_i\setminus F_j \text{ for } j=1,\ldots,{i-1}\}$$ and where $e_L=\sum_{j\in L}e_{j}$ for each ... | 2,833 | 2,154 | 2,448 | 2,579 | 4,024 | 0.768584 | github_plus_top10pct_by_avg |
0.733
WBC (10^9^/mL) 6.2 ± 5.7 5.7 ± 1.4 0.560 4.9 ± 1.2 6.0 ± 3.3 0.117
PLT (10^9^/mL) 170.2 ± 58.4 19... | 2,834 | 5,285 | 1,976 | 2,182 | null | null | github_plus_top10pct_by_avg |
nu}\Box R
\quad\; , \quad
\mathcal{M}_3=R^{\mu\nu}\Box^2R_{\mu\nu}
\\~
\\
\mathcal{M}_5=\nabla^{\mu}\Box R \nabla_{\mu} R
\;\;\, , \;\;
\mathcal{M}_6=\nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}R \nabla^{\mu} R^{\nu\alpha}
\;\;\, , \;\;
\mathcal{M}_{10}=\big(\Box R \big)^2
\;\; \, , \;\;
\mathcal... | 2,835 | 2,831 | 2,370 | 2,549 | null | null | github_plus_top10pct_by_avg |
gcd}(g_k^*, x^{m_i}-1)=1$.
\[\]
(v) ${\mathcal R}_i=\bigoplus_{k=1}^s{\mathcal R}_{ij}$.
\[\]
(vi) For each $k=1,2,\ldots,s$, the mapping $\phi_{ik}:~{\mathcal R}_{ik}\rightarrow R/(g_k^{*d_{ik}})$, defined by $$\phi_{ik}:~fb_k\widehat{g}_k^*+(x^{m_i}-1)\mapsto f+(g_k^{*d_{ik}}), ~\mbox{where}\; f\in R,$$ is a wel... | 2,836 | 1,430 | 1,946 | 2,591 | 4,108 | 0.768053 | github_plus_top10pct_by_avg |
ay". If you split a small vim window but still with a long message bar, the hit-enter prompt won't occur.
A:
Is there some way to totally avoid hit-enter prompts that are caused only by "not enough message space"?
No. Only "more prompts" can be fully disabled. See :help hit-enter.
I believe the hit-enter prompt her... | 2,837 | 670 | 1,944 | 2,269 | 51 | 0.831458 | github_plus_top10pct_by_avg |
sible.
[*(iv)*]{} Suppose that $m(P)=2$ and $m(Q)=1$. Then $P = P_r + P_p$ and $Q=Q_q$. By Lemma \[a20Sep16\] the equality $[P,Q]=1$ implies that $[P_p, Q_q]=0$ and $[P_r, Q_q]=1$. Then, $q\geq 0$, by Lemma \[a20Sep16\]. The case $q=0$ is not possible since then both $P_r, Q_q\in K[H]$ and this would contradict the eq... | 2,838 | 2,147 | 2,667 | 2,551 | 4,021 | 0.768608 | github_plus_top10pct_by_avg |
dependent set of data). As in the mentioned papers, expressions such as (\[abr1\]) will be referred to here as ‘ideal’ estimators.
It was once believed that $f_t$ in (\[abr1\]) could be replaced by $f$, but Terrell and Scott (1992) showed that in this case the bias reduction at a single $t$ depends heavily on the tail... | 2,839 | 2,851 | 2,687 | 2,660 | 1,931 | 0.784305 | github_plus_top10pct_by_avg |
ith the norm $(4)$ (so that it is isometric to $H(1)$ by Theorem \[210\]). Then by observing a formal matrix description of an element of $\underline{M}(R)$ for a $\kappa$-algebra $R$, explained in Section \[m\], the above formal matrix turns to be $$\begin{pmatrix}1-2z_j&0&0&0\\\pi z_j&1&0&\pi z_j\\\pi z_j&0&1&0\\\pi ... | 2,840 | 1,509 | 2,530 | 2,682 | null | null | github_plus_top10pct_by_avg |
an. J. Sci. Technol. **36**(A3) (2012) (Special Issue-Mathematics), 371-376.
M. Mursaleen, A.K. Noman, Applications of Hausdorff measure of noncompactness in the spaces of generalized means, Math. Inequal. Appl. **16** (2013) 207-220.
M. Stieglitz, H. Tietz, Matrix transformationen von folgenräumen eine ergebnisübers... | 2,841 | 1,751 | 566 | 2,701 | 3,041 | 0.775282 | github_plus_top10pct_by_avg |
plane. Overall, the numerical results are exceedingly close to the analytic potential, with the mean relative error less than 0.1%. The errors are largest near the sphere boundary whose exact shape is not well resolved by any of the adopted grids.
[cccccc]{}\[!t\] Cartesian & $[-0.5,0.5]\times[-0.5,0.5]\times[-0.5,0.5... | 2,842 | 584 | 3,043 | 2,801 | null | null | github_plus_top10pct_by_avg |
tations by computing adversarial samples for a CNN. In particular, influence functions [@CNNInfluence] were proposed to compute adversarial samples, provide plausible ways to create training samples to attack the learning of CNNs, fix the training set, and further debug representations of a CNN. [@banditUnknown] discov... | 2,843 | 909 | 3,238 | 2,576 | null | null | github_plus_top10pct_by_avg |
anisotropic power spectra.
Finally, putting eq. (\[distortion\]) into eq. (\[projection\]), it can be seen that the one-dimensional redshift-space power spectrum is related to the isotropic three-dimensional real-space power spectrum by a linear integral equation: $$P (k_{\parallel}) = \int_{k_{\parallel}}^\infty
W(k... | 2,844 | 4,284 | 2,910 | 2,665 | 3,498 | 0.771923 | github_plus_top10pct_by_avg |
---------------
The observable matter states in heterotic–string vacuum with $(2,2)$ world–sheet supersymmetry is embedded in the $\bf{27}$ representation of $E_6$. In the free fermionic construction that we adopt here, and using the basis vectors in (\[421\]), the $E_6$ is first broken to the $SO(10)\times U(1)$ symm... | 2,845 | 1,105 | 1,777 | 2,866 | 3,134 | 0.774625 | github_plus_top10pct_by_avg |
15 Continuous HeartMate II RIFLE criteria ... | 2,846 | 4,913 | 2,104 | 2,044 | null | null | github_plus_top10pct_by_avg |
s exponentially with the chain order and the available data is too sparse for proper parameter inferences. Thus, we show further evidence that the memoryless model seems to be a quite practical and legitimate model for human navigation on a page level.
- By abstracting away from the page level to a topical level, th... | 2,847 | 2,061 | 3,073 | 2,031 | null | null | github_plus_top10pct_by_avg |
sition \[magneticeigenbladeterminant\], below, yields $f_1\equiv f_2\equiv0$, which gives $\ker\left({\mathbf{N}}_2\right)=\{0\}$. $\blacksquare$
Now we want to determine the prefactor in Equation . Recall that the determinant of a diagonalizable operator is defined as the product of its eigenvalues, if it exists. We ... | 2,848 | 3,823 | 2,434 | 2,566 | null | null | github_plus_top10pct_by_avg |
bination of semistandard homomorphisms using three applications of Lemma \[lemma7\], from which it follows that ${\hat\Theta_{E}}\neq0$.
To prove that ${\hat\Theta_{T}}\circ{\hat\Theta_{C}}={\hat\Theta_{E}}$, use the notation of Proposition \[tabcomp\], with $S=C$. Suppose $X\in\calx$ is such that the coefficient of $... | 2,849 | 1,327 | 1,166 | 2,669 | 3,940 | 0.769125 | github_plus_top10pct_by_avg |
named entities, geographical locations and temporal expressions. What would be the most descriptive mathematical models for each of these semantic annotations?
**Similarity Functions**. Given a pair of named entities, geographical locations or temporal expressions; how can we efficiently compute the similarity between... | 2,850 | 182 | 1,337 | 2,689 | 2,168 | 0.782061 | github_plus_top10pct_by_avg |
Hida distribution, see Theorem \[magnetictheorem\].
- The results in Theorem \[magnetictheorem\] provide us with the generating functional for a charged particle in a constant magnetic field.
- The generalized expectations (generating functional at zero) yields the Greens functions to the corresponding Schrödinge... | 2,851 | 1,023 | 2,525 | 2,704 | 3,538 | 0.771691 | github_plus_top10pct_by_avg |
he state of the system and the eigenmode amplitudes of Ref. [@Terry2006] can be recovered. Consequently, the governing Eq. can be manipulated to derive nonlinear equations that describe the evolution of the eigenmode amplitudes and their interactions. The method relies on the jump conditions given in Eq. . Since the j... | 2,852 | 3,598 | 3,105 | 2,715 | null | null | github_plus_top10pct_by_avg |
1^{+0.049}_{-0.048}$ from [@WMAP]. We also calculate $\Omega_{\hbox{\scriptsize asymp}}$, the fractional density of matter that *cannot* be determined through gravity, to be $0.197_{\pm 0.017}$, which is nearly equal to the fractional density of nonbaryonic matter $\Omega_m-\Omega_{B} =
0.196^{+0.025}_{-0.026}$ [@WMAP]... | 2,853 | 3,793 | 2,798 | 2,453 | 2,461 | 0.779565 | github_plus_top10pct_by_avg |
t ${\mathrm{Hom}}(a,b)$ consists of the triples $(b,f,a)$, where $$f={\sigma }_{i_n}^{{r}_{i_{n-1}}\cdots {r}_{i_1}(a)}\cdots
{\sigma }_{i_2}^{{r}_{i_1}(a)}{\sigma }_{i_1}^a$$ and $b={r}_{i_n}\cdots {r}_{i_2}{r}_{i_1}(a)$ for some $n\in {\mathbb{N}}_0$ and $i_1,\ldots ,i_n\in I$. The composition is induced by the g... | 2,854 | 2,508 | 2,568 | 2,545 | 4,178 | 0.767537 | github_plus_top10pct_by_avg |
t $O$ randomly: $$\begin{gathered}
{{\mathbb P}}\big(\{\theta(i,j)\in A\}\cap \{M\geq 2\} \cap \Omega(i,j)\big)= \sum_{m=2}^5 {{\mathbb P}}\big(\{\theta(i,j)\in A\}\cap \{M=m\}\cap \Omega(i,j)\big)\\
\begin{aligned}
\leq & {{\mathbb P}}\big(\bigcup_{i\not= j\in \{1,\dots,m\}}\{\theta(\underline{i},\underline{j})\in A... | 2,855 | 1,390 | 2,300 | 2,649 | null | null | github_plus_top10pct_by_avg |
d], $$\begin{aligned}
\delta_{K\pi} &= \mathrm{arg}\left(- \frac{1-\frac{1}{2} \tilde{t}_1 }{1+\frac{1}{2} \tilde{t}_1 } \right)
= -\mathrm{Im} (\tilde{t}_1)\,, \label{eq:strongphase} \end{aligned}$$ where in the last step we neglect terms of relative order of $\varepsilon^2$.
After that we can determine $\tild... | 2,856 | 4,357 | 2,851 | 2,471 | null | null | github_plus_top10pct_by_avg |
of $S$ always has a positive lower bound in probability. Combing these three facts, it gives $$\|\lambda\|[ \theta^\top S_K\theta + O_{p}(K^{-1/2}) o_{p}(K^{1/2}) ] = O_{p}(K^{-1/2}).$$ So, we have $$\|\lambda\|= O_{p}(K^{-1/2}).$$ Furthermore, $$\label{eqA5}
\max_{1\leq k\leq K}|U_{km}|= O_{p}(K^{-1/2})o_{p}(K^{-1/... | 2,857 | 3,716 | 2,501 | 2,641 | null | null | github_plus_top10pct_by_avg |
M{l}$, their conjugate momenta $\MM{\pi}$ and the conjugate momentum $(\phi)$ to the advected quantities $(a)$ may be eliminated from equations (\[EL advected 1\]-\[EL advected 5\]) to obtain the weak form of the Euler-Poincaré equation with advected quantities: $${\frac{\partial }{\partial t}}{\frac{\delta \ell}{\delt... | 2,858 | 4,322 | 1,868 | 2,242 | null | null | github_plus_top10pct_by_avg |
on}\gamma^\mu\lambda&=&
-\overline{\lambda}\gamma^\mu\epsilon&=&
-\left(\overline{\epsilon}\gamma^\mu\lambda\right)^\dagger\ ,\\
\overline{\epsilon}\gamma^\mu\gamma_5\lambda&=&
\overline{\lambda}\gamma^\mu\gamma_5\epsilon&=&
\left(\overline{\epsilon}\gamma^\mu\gamma_5\lambda\right)^\dagger\ ,\\
\overline{\epsilon}\gamm... | 2,859 | 3,186 | 3,158 | 2,761 | null | null | github_plus_top10pct_by_avg |
e consider as a dimension vector of $\Gamma$. Define $\varphihat'\in{\rm Rep}_{\Gamma,\v'}(\K)$ as the restriction of $\varphihat$ to $\Vhat'$. It is a non-zero subrepresentation of $\varphihat$. It is now possible to define a graded vector subspace $\Vhat''=\bigoplus_{i\in I}V_i''$ of $\Vhat$ such that the restriction... | 2,860 | 1,637 | 2,156 | 2,588 | null | null | github_plus_top10pct_by_avg |
{}, we get the following abstract error estimation:
Assume [**[I1]{}**]{}-[**[I3]{}**]{} hold, $C$ be a given constant in $(0,1)$ and $\alpha \ge \frac{(1+\delta)^2C_1}{4(1-C)^2}$. Let $u$ and $u_h$ be the solution to equations (\[eq:ellipticeq\]) and (\[eq:dg\]), respectively. Then $$\3bar u-u_h\3bar \lesssim C_A\big... | 2,861 | 1,148 | 1,686 | 2,810 | null | null | github_plus_top10pct_by_avg |
ongleftrightarrow}}}x
\text{ from $v$ such that }v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}{\underline{b}}\}.\end{aligned}$$ On the event $\{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}... | 2,862 | 1,108 | 1,403 | 2,832 | null | null | github_plus_top10pct_by_avg |
2}=4m_0^2$ with $n=n^{\prime}=0$.](prl1.eps){width="3in"}
It is easy to show that this function has a maximum near the threshold. If we consider $\omega$ near $2m_0$, the function $\mu_{\gamma}^{(2)}=f(X)$, where $X=\sqrt{4m_0^2-\omega^{2}}$ has a maximum for $X= {\pi\phi_{00}^{(2)}/m_0}^{1/3}$, which is very close to... | 2,863 | 3,208 | 3,159 | 2,844 | 3,853 | 0.769688 | github_plus_top10pct_by_avg |
nd eigenvectors determined here with a 30 state expansion for each $\theta$-parity. The ordering convention adopted for the states was taken as $$\notag \Psi^{+}_{0,-2}, \Psi^{+}_{0,-1},...\Psi^{+}_{5,2} ,
\Psi^{-}_{1,-2}...\Psi^{-}_{6,2}$$ yielding a Hamiltonian matrix blocked schematically into
$$\left( \begin{array... | 2,864 | 3,493 | 3,009 | 2,714 | null | null | github_plus_top10pct_by_avg |
ether $\phi$ lifts to a homomorphism $$\psi: \: \pi_1\left( [T^4/{\mathbb Z}_2] \right) \: \longrightarrow \:
{\mathbb Z}_4.$$ First note $$\pi_1\left( [T^4/{\mathbb Z}_2] \right) \: = \: {\mathbb Z}_2 \rtimes
{\mathbb Z}^4,$$ where the nontrivial element in ${\mathbb Z}_2$ acts as multiplication by $-1$ on ${\... | 2,865 | 2,204 | 2,832 | 2,490 | null | null | github_plus_top10pct_by_avg |
====================================================================
To extend this method to more general equations with advected quantities is very simple: take the Lagrangian obtained from equation (\[inverse map sec\]) and add variables to represent higher-order derivatives. For the sake of brevity we shall comput... | 2,866 | 1,687 | 2,758 | 2,624 | null | null | github_plus_top10pct_by_avg |
ay obligate the person in two tefillos, whereas crossing the
dateline in a westward direction (effectively stepping into the next
day without nightfall) does not require a new tefilloh. These two
views regarding tefilloh are expounded upon in Rav Betzalel Stern’s
Betzel Hachochmo, and Rav Yechezkel Roth’s Emek ... | 2,867 | 5,212 | 3,165 | 1,842 | 627 | 0.803665 | github_plus_top10pct_by_avg |
s}}}}$, with $\eta$ the viscosity of the solvent. Here $\left\langle \triangle r_{\textrm{cls}}^{2}(n)\right\rangle $ is the mean square displacement of the clusters after $n$ cycles, defined as $$\left\langle \triangle r_{\textrm{cls}}^{2}(n)\right\rangle =\dfrac{1}{N_{n_{c}}}{ \sum_{i=1}^{N_{n_{c}}}\triangle\mathbf{r... | 2,868 | 3,986 | 3,818 | 2,899 | 3,105 | 0.774858 | github_plus_top10pct_by_avg |
thbf x},{\mathbf x}')=k({\boldsymbol{\mathrm{r}}})$ where ${\boldsymbol{\mathrm{r}}}={\mathbf x}-{\mathbf x}'$, so the covariance only depends on the distance between the input points. In that case we can also work with the spectral density, which is the Fourier transform of the stationary covariance function $$\label{... | 2,869 | 1,447 | 1,900 | 2,709 | null | null | github_plus_top10pct_by_avg |
ental representations is related to the existence of two SU(N)-invariant tensors, namely the Kronecker symbols ${\delta^\alpha}_\beta$ and ${\delta_\alpha}^\beta$ and the totally antisymmetric symbols $\epsilon^{\alpha_1\cdots\alpha_N}$ and $\epsilon_{\alpha_1\cdots\alpha_N}$, which themselves are directly connected to... | 2,870 | 2,795 | 2,726 | 2,731 | null | null | github_plus_top10pct_by_avg |
is $$\begin{aligned}
S=\int dt\left(-m\sqrt{V}\sqrt{-G_{tt}-G_{\rho\rho}\dot\rho^2}-\frac{qQ}{2\rho^2}\right)\;.\end{aligned}$$ The Hamiltonian is $$\begin{aligned}
H&=\frac{-m\sqrt{V}G_{tt}}{\sqrt{-G_{tt}-G_{\rho\rho}\dot\rho^2}}+\frac{qQ}{2\rho^2}\nonumber\\
&=\sqrt{-G_{tt}(G^{\rho\rho}\pi^2+m^2V)}+\frac{qQ}{2\rho^2... | 2,871 | 4,059 | 2,525 | 2,719 | null | null | github_plus_top10pct_by_avg |
s' movement inside the bus is part of pre-movement time) was large, the first person leaves the bus 37 seconds after initialization of the alarm system (*t* = 37*s*), see [Fig 4a](#pone.0201732.g004){ref-type="fig"}. Process of path selection during experiment 1 is presented in [Fig 4](#pone.0201732.g004){ref-type="fig... | 2,872 | 1,404 | 1,694 | 2,444 | null | null | github_plus_top10pct_by_avg |
}$ left translates of $A^2\cap N$. \[lem:slicing\] implies that $A^2\cap N$ is a $K^3$-approximate group, and so the existence of $P_1$ follows from \[cor:ruzsa\] and the existence of $P_2$ follows from \[cor:chang.ag\].
In the special case in which $\Bbbk={\mathbb{C}}$, an argument of Breuillard and Green shows that ... | 2,873 | 2,005 | 1,942 | 2,765 | 2,159 | 0.782153 | github_plus_top10pct_by_avg |
ype 2 Equivalence.** If $(f_{k})$ are functions on $J$ that are integrable on every interior subinterval, then the following are equivalent statements.
\(a) For every interior subinterval $I$ of $J$ there is an integer $m_{I}\geq0$, and hence a smallest integer $m\geq0$, such that certain indefinite integrals $f_{k}^{... | 2,874 | 3,571 | 3,364 | 2,675 | 3,939 | 0.769132 | github_plus_top10pct_by_avg |
(http://88t.eu/Pictures/sh/1/intr_17.jpg) no-repeat center center;
}
.pic-18 {
animation-delay: 102s;
-o-animation-delay: 102s;
-moz--animation-delay: 102s;
-webkit-animation-delay: 102s;
background: url(http://88t.eu/Pictures/sh/1/intr_18.jpg) no-repeat center center;
}
.p... | 2,875 | 149 | 3,164 | 2,906 | 97 | 0.826455 | github_plus_top10pct_by_avg |
\int\,d\eta\,d\eta^\dagger\,\eta^\dagger=0\ \ ,\ \
\int\,d\eta\,d\eta^\dagger\,\eta^\dagger\eta=1\ \ ,\ \$$ while the result for any linear combination of these $\eta$-monomials is given by the appropriate linear combination of the resulting integrations (the usual integral over Grassmann even variables being also lin... | 2,876 | 1,571 | 2,492 | 2,638 | null | null | github_plus_top10pct_by_avg |
emitters
=============================================
{width="40.00000%"}
In addition to characterizing fully coherent radiation, CL polarimetry allows us to determine whether the measured radiation contains an unpolarized contribution such as in the case of incoherent luminescence from bulk or nan... | 2,877 | 651 | 3,408 | 2,972 | null | null | github_plus_top10pct_by_avg |
y. $CFT_2$ also belongs to this case.
The modular group is generated by the $T$ and $S$ transformations. The T-transformation leads to $$(\alpha,\beta)\rightarrow (\alpha,\beta+\alpha), \hs{3ex}
T=\left( \ba{cc}
1&0\\
1&1
\ea
\right).$$ The $S$-transformation leads to $$(\alpha,\beta)\rightarrow (-\beta,\alpha),\hs{3e... | 2,878 | 1,204 | 2,445 | 2,658 | null | null | github_plus_top10pct_by_avg |
e $$a^mb^n=(a^{p+m+n}b^m)^{-1}(a^{p+m+n}b^n).$$It is clear that $a^{p+m+n}b^m,a^{p+m+n}b^n \in S$.
\[twostraightl\] If $S=F_{D}\cup \widehat{F}\cup \widehat{\Lambda}_{I,p,d}\cup \Sigma_{p,d,P}$ is a left I-order in $\mathcal{B}$, then it is straight.
From corollaries \[twostraightr\] and \[twostraightl\], we have t... | 2,879 | 615 | 3,083 | 2,890 | null | null | github_plus_top10pct_by_avg |
105 (50.5%)
Secondary eye
Total **75 (100%)**
1---diagnosed before the study period 48 (64.0%)
2---diagnosed after the first study eye 15 (20.0%)
3---both eyes diagnosed at same time 12 (16.0%)
######
Description ... | 2,880 | 6,317 | 1,137 | 1,339 | null | null | github_plus_top10pct_by_avg |
ng a stronger iteration axiom (but not a larger large cardinal).
With the conclusion of [@T3] restored, [@T4], [@LT2], and [@T] are re-instated. We shall then proceed to improve the results of the two latter ones.
PFA$(S)[S]$ and the role of $\omega_1$
======================================
*PFA$(S)$* is the Proper ... | 2,881 | 2,307 | 3,219 | 2,886 | 1,874 | 0.784817 | github_plus_top10pct_by_avg |
this to a countable dense family of $f$, it follows that $\mu^{(n)} \to \gamma_{\mathbb{C}}$ weakly almost surely.
In the general case, since ${\left\vert G^{(n)} \right\vert} \to \infty$, each subsequence of $\mu^{(n)}$ has a subsequence $\mu^{(n_j)}$ for which, say, ${\left\vert G^{(n_j)} \right\vert} \ge j$, so tha... | 2,882 | 3,505 | 2,777 | 2,447 | 4,136 | 0.767846 | github_plus_top10pct_by_avg |
s the closure of this core, which in this case of the topology being induced by the filterbase, is just the core itself. $A_{1}$ by its very definition, is a positively invariant set as any sequence of graphs converging to **$\textrm{Atr}(A_{1})$ must be eventually in $A_{1}$: the entire sequence therefore lies in $A_{... | 2,883 | 2,149 | 3,144 | 2,692 | 2,082 | 0.782936 | github_plus_top10pct_by_avg |
',\omega,E',E)\in S^2\times I^2\ |\ \omega'\cdot\omega-\mu_{11}(E',E)=0\},$$ then we would have (assuming that pertinent functions are Borel integrable) \[k-11\] (K\_[11]{})(x,,E)=\_[S’I’]{} \_[11]{}(x,’,,E’,E)(x,’,E’)d(’)dE’. This expression has the pleasant feature that the differential cross section $\underline{\sig... | 2,884 | 846 | 2,637 | 2,804 | null | null | github_plus_top10pct_by_avg |
t $R_{L_\lambda}^G(1)(x)=|P_\lambda|^{-1}\#\{g\in G\,|\, g^{-1}xg\in P_\lambda\}$, hence $R_{\mathfrak{l}_\lambda}^{\mathfrak{g}}(1)$ is the Lie algebra analogue of $R_{L_\lambda}^G(1)$ and the two functions take the same values on elements of same type.\[Rl=RL\]
We have $$\calF^\mathfrak{g}\left(Q_{\mathfrak{l}_\lamb... | 2,885 | 1,566 | 2,115 | 2,562 | null | null | github_plus_top10pct_by_avg |
--C80 114.1 (5)
C14---C19---H19 119.8 O2---C79---N2 125.1 (6)
C18---C19---H19 119.8 O2---C79---H79 117.4
C21---C20---C25 118.8 (4) N2---C79---H79 117.4
C21---C20---P6 123.1 (4) N2---C80---H80A 109.5
C25---C20---P6 117.8 (4) N2---C80---... | 2,886 | 539 | 1,999 | 2,628 | null | null | github_plus_top10pct_by_avg |
3 - 5*m**2*p**3 - m*p**3 + 3*m wrt m?
-1620*p**3 - 18
What is the second derivative of 24*k**5*u**2 + 53*k**4*u**3 - 2301*k*u**3 wrt k?
480*k**3*u**2 + 636*k**2*u**3
What is the third derivative of -1271*f*n**4*s + f*n**3*s - f*n**3 + f*n**2*s - 5*f*n**2 - 3*n*s - 24 wrt n?
-30504*f*n*s + 6*f*s - 6*f
Find the second de... | 2,887 | 2,679 | 3,074 | 2,984 | null | null | github_plus_top10pct_by_avg |
u _{n}(dx)=f_{n}(x)dx,$ $$\limsup_{n}d_{k}(\mu ,\mu _{n})\times \theta ^{\rho _{h}+\varepsilon
}(n)<\infty \label{reg10}$$for some $\varepsilon >0.$ Then $\mu (dx)=f(x)dx$ with $f\in W^{q,p}.$
Moreover, for $\delta ,\varepsilon >0$ and $n_{\ast }\in {\mathbb{N}}$, let $$\begin{aligned}
A(\delta )& =\left\vert \mu \ri... | 2,888 | 1,145 | 1,224 | 2,947 | null | null | github_plus_top10pct_by_avg |
bf{w}}),$$ and hence $$D_{\mathbf{u}}D_{\mathbf{v}}h({\mathbf{w}}) \geq \min\{ D_{\mathbf{u}}^2 h({\mathbf{w}}), D_{\mathbf{v}}^2 h({\mathbf{w}})\}.$$
By continuity we may assume ${\mathbf{u}}, {\mathbf{v}}, {\mathbf{w}}\in \Lambda_{++}$. Then the polynomial $$\begin{aligned}
& g(x,y,z):=h(x{\mathbf{u}}+y{\mathbf{v}}+... | 2,889 | 2,561 | 2,819 | 2,749 | null | null | github_plus_top10pct_by_avg |
iscretized only at the final stage is fundamentally different from the method employed in the original study by [@ld08] in which the optimization problem was solved in a fully discrete setting (the two approaches are referred to as “optimize-then-discretize” and “discretize-then-optimize”, respectively, cf. [@g03]). A ... | 2,890 | 1,815 | 1,056 | 2,775 | 4,091 | 0.768174 | github_plus_top10pct_by_avg |
al{G}^{1/3}}+
\frac{\mathcal{G}^{1/3}}{k_\perp^{\prime\prime
2}-k_{\perp}^{\prime 2}}\right] \label{fi}$$ where $\mathcal{G} =6 \pi\sqrt{3}D-\Lambda^{*3}+54 \pi^2
\phi_{n,n^{\prime}}^{(i) 2}(k_\perp^{\prime\prime
2}-k_\perp^{\prime 2})^2$ with $$D=\sqrt{-(k_\perp^{\prime 2}-k_\perp^{\prime\prime
2})^2\Lambda^{*3}\phi_{... | 2,891 | 2,140 | 3,069 | 2,756 | null | null | github_plus_top10pct_by_avg |
ince $3\ls v\ls a-1$, we can take $\delta=\sigma$.
If $a\equiv0$, take $\gamma={\hat\Theta_{A}}+{\hat\Theta_{B}}$. By Proposition \[abhoms\], $\gamma$ is a homomorphism from $S^\mu$ to $S^\la$. By Lemma \[uab\] and Lemma \[countu\], $$\delta\circ\gamma=\mbinom{u-v}{a-v}\mbinom{u-a}2{\hat\Theta_{D}}.$$ The first term i... | 2,892 | 1,169 | 1,824 | 2,784 | 3,873 | 0.769612 | github_plus_top10pct_by_avg |
M_\pi(C^{\lambda},k)=\cM_\pi(C^{\lambda+\varepsilon},k)$. We claim one can find a projective curve $\cD\subset \PP^2_w$, $w=(w_0,w_1,w_2)$ as in in the proof of Lemma \[lemma:global-realization\], where $\deg_w\cD=c(1+\Delta w_2)$ for $c>0$, $c\equiv [C] \mod (w_2)$ and a big enough $\Delta\gg 0$ and $a\equiv k \mod (... | 2,893 | 2,074 | 2,065 | 2,626 | null | null | github_plus_top10pct_by_avg |
re indices $A\neq B$, which means that real wave vectors are excluded from our consideration (we do not consider here the mixed case for wave vectors involving real and complex wave vectors). Under the above hypotheses, the $k$ wave vectors (\[eq:wv\]) and their complex conjugates $$\bar{{\lambda}}^A(u)={\left( \bar{{\... | 2,894 | 965 | 3,248 | 2,998 | null | null | github_plus_top10pct_by_avg |
\tilde L^{n-4k}$ is obtained by gluing the manifold $\tilde L^{n-4k}_{x} \cup \tilde L^{n-4k}_y$ with the manifold $\tilde L^{n-4k}_z$ along the common boundary $\tilde
\Lambda^{n-4k-1}$. Note that the group of the framing of the last manifold $\tilde \Lambda^{n-4k-1}_z$ is the subgroup $\I_3 \subset
\Z/2 \int \D_4$.
... | 2,895 | 2,519 | 2,009 | 2,750 | 3,626 | 0.771186 | github_plus_top10pct_by_avg |
@MR2002d:14084]; we reproduce the list of curves obtained in [@MR2002d:14084] in an appendix at the end of this paper (§\[appendix\]). For another classification, from a somewhat different viewpoint, we refer to [@MR1698902]. For these curves, the limits can be determined using the results in [@MR2002d:14083] (see also... | 2,896 | 1,270 | 2,469 | 2,802 | null | null | github_plus_top10pct_by_avg |
_{m,m^\prime} \delta_{h,h^\prime} \delta_{k,k^\prime}
\,.
\label{eq:orthogonal-relation}$$ Here the overbar denotes complex conjugation, and the volume element is given by $$\int_{\Sigma_u} {\mathrm{dVol}\,}= \lim_{T\rightarrow \infty}
\int_{-T}^T\text{d}\tau \int_0^{2\pi}\text{d}\varphi
\int_0^{\pi}\text{d}\psi \sqrt{... | 2,897 | 2,773 | 2,257 | 2,702 | null | null | github_plus_top10pct_by_avg |
pchi}$$ for all $i,j\in I\setminus \{p\}$. It is a small exercise to check that then $({\sigma }_p^\chi )^*\chi $ is $p$-finite, and $$\begin{aligned}
\label{eq:rp2}
c_{pj}^{r_p(\chi )}=c_{pj}^\chi \quad \text{for all $j\in I$},
\qquad r_p^2(\chi )=\chi .\end{aligned}$$ The reflections $r_p$, $p\in I$, generate a s... | 2,898 | 2,275 | 2,553 | 2,652 | null | null | github_plus_top10pct_by_avg |
${\prod{\Phi}} = (\thinspace{\prod{\Psi}}) \mid R$. By Theorem \[T:CHC\_RSTR\_EQ\_RSTR\_CHC\], the restriction of the choice space equals the choice space of the restriction: $(\thinspace\prod\Psi) \mid R = \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$. Transitivity of equality implies ${\prod{\Phi}} = \prod ({{\Psi... | 2,899 | 1,593 | 1,944 | 2,780 | null | null | github_plus_top10pct_by_avg |
explain how to compute $X\mapsto \sigma(m)^t\cdot h \cdot X+\sigma(X)^t\cdot h \cdot m$ explicitly. Recall that for a $\kappa$-algebra $R$, we denote an element $m$ of $\underline{M}(R)$ by $(m_{i,j}, s_i\cdots w_i)$ with a formal matrix interpretation $m=(\pi^{max\{0, j-i\}}m_{i,j}) \mathrm{~together~with~}z_i^{\ast... | 2,900 | 1,094 | 1,484 | 2,987 | 3,967 | 0.768911 | github_plus_top10pct_by_avg |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.